Philosophy of Mathematics

Edited by Øystein Linnebo (Birkbeck College)
Assistant editor: Sam Roberts (Birkbeck College, University of Oslo)
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  1. David Hilbert’s ’Vorlesungen’ Logic and Foundations of Mathematics.Vito Michele Abrusci - 1989 - In G. Corsi, C. Mangione & M. Mugnai (eds.), Atti Del Convegno Internazionale di Storia Della Logica, San Gimignano, 1987. Editrice Cooperativa Libraria Universitaria Editrice, 1989. pp. 333-338..
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    Epistemology of Mathematics
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    Set Theory
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  2. ¿ES LA MATEMÁTICA LA NOMOGONÍA DE LA CONCIENCIA? REFLEXIONES ACERCA DEL ORIGEN DE LA CONCIENCIA Y EL PLATONISMO MATEMÁTICO DE ROGER PENROSE / Is Mathematics the “nomogony” of Consciousness? Reflections on the origin of consciousness and mathematical Platonism of Roger Penrose.Miguel Acosta - 2016 - Naturaleza y Libertad. Revista de Estudios Interdisciplinares 7:15-39.
    Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...)
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  3. Ce Que les Mathématiques Ont Apporté À la Philosophie Contemporaine.Evandro Agazzi - 1981 - Epistemologia 4 (1):5.
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  4. Non-contradiction et existence en mathématique.Evandro Agazzi - 1978 - Logique Et Analyse 21 (84):459.
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    Ontology of Mathematics
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     Mathematical Cognition, Misc
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    Mathematical Truth
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  5. Ausgewählte Texte Lateinisch-Deutsch.Albert Albertus & Fries - 1981
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     Indeterminacy in Mathematics
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    Mathematical Truth
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     Axiomatic Truth
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  6. On the Philosophical Adequacy of Set Theories.E. Alchourrón Carlos - 1987 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 2 (2-3):567-574.
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     Phenomenology of Mathematics
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    Mathematical Truth
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  7. Priorities and Responsibilities in the Reform of Mathematical Education.Alexander Wittenberg Alexander Wittenberg - 1968 - Dialectica 22 (1):58-74.
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  8. Arithmetical Algorithms for Elementary Patterns.Samuel A. Alexander - 2015 - Archive for Mathematical Logic 54 (1-2):113-132.
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  9. Identity and Intensionality in Univalent Foundations and Philosophy.Angere Staffan - forthcoming - Synthese:1-41.
    The Univalent Foundations project constitutes what is arguably the most serious challenge to set-theoretic foundations of mathematics since intuitionism. Like intuitionism, it differs both in its philosophical motivations and its mathematical-logical apparatus. In this paper we will focus on one such difference: Univalent Foundations’ reliance on an intensional rather than extensional logic, through its use of intensional Martin-Löf type theory. To this, UF adds what may be regarded as certain extensionality principles, although it is not immediately clear how these principles (...)
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  10. Mathematics And Value.W. Anglin - 1991 - Philosophia Mathematica 2:145-173.
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  11. On Formally Measuring and Eliminating Extraneous Notions in Proofs.Andrew Arana - 2009 - Philosophia Mathematica 17 (2):189-207.
    Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
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  12. Arguments and Elements of Realistic Interpretation of Mathematics: Arithmetical Component.E. I. Arepiev & V. V. Moroz - 2015 - Liberal Arts in Russia 4 (3):198-204.
    The prospects for realistic interpretation of the nature of initial mathematical truths and objects are considered in the article. The arguments of realism, reasons impeding its recognition among philosophers of mathematics as well as the ways to eliminate these reasons are discussed. It is proven that the absence of acceptable ontological interpretation of mathematical realism is the main obstacle to its recognition. This paper explicates the introductory positions of this interpretation and presents a realistic interpretation of the arithmetical component of (...)
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  13. The nature of numbers in the light of a broader interpretation of reality.I. E. Arepiev - 2014 - Liberal Arts in Russia 3 (4):229--236.
    The article concerns the problems of philosophy of mathematics. The traditional problem of the existence of mathematical objects and truths is solved through the reconstruction of the ontological concepts of the real and possible. The work deals with the problems and development options of certain trends in the philosophy of mathematics. The article describes the arguments in favour of a realistic interpretation of the foundations of mathematics in the light of the extended interpretation of reality. The work contains also the (...)
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  14. Jody Azzouni. Tracking Reason: Proof, Consequence and Truth: Critical Studies/Book Reviews.Asmus Conrad - 2009 - Philosophia Mathematica 17 (3):369-377.
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  15. Problema Intuitsii V Filosofii I Matematike.Valentin Ferdinandovich Asmus - 1963 - Izd-Vo Sots.-Ekon. Lit-Ry.
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  16. Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuumt.Atten Mark, Dalen Dirk & Tieszen Richard - 2002 - Philosophia Mathematica 10 (2):203-226.
    Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...)
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  17. The Role of Mathematics in the Exploration of Reality.Karl Egil Aubert - 1982 - Inquiry 25 (3):353 – 359.
    In his well?known paper from 1954, Herbert A. Simon sets out to demonstrate that it is possible, in principle, to make public predictions within the social sciences that will be confirmed by the events. However, Simon's proof by means of the Brouwer fixed?point theorem not only rests on an illegitimate use of continuous variables, it is also founded on the questionable assumption that facts ? even on the level of possibilities ? can be established by purely mathematical means. The ?proof? (...)
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  18. Méthode axiomatique et négation chez Hilbert.Eric Audureau - 2007 - Philosophia Scientiae 11 (2):67-96.
    a) The epistemology advocated by Hilbert through the development of proof theory is already held in his Paris Address in 1900.b) The application of the fondamental principle of Hilbert’s epistemology to the characterization of logical negation is one of the main problems of Hilbert’s proof theoryc) In order to characterize negation a property of signs one has to give up the axiomatic method, namely, the core of Hilbert’s epistemology.Résuméa) La doctrine de la connaissance défendue par Hilbert au cours du développement (...)
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  19. By Dennis E. Hesseling.Jeremy Avigad - unknown
    The early twentieth century was a lively time for the foundations of mathematics. This ensuing debates were, in large part, a reaction to the settheoretic and nonconstructive methods that had begun making their way into mathematical practice around the turn of the twentieth century. The controversy was exacerbated by the discovery that overly na¨ıve formulations of the fundamental principles governing the use of sets could result in contradictions. Many of the leading mathematicians of the day, including Hilbert, Henri Poincar´e, ´.
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  20. Dedekind's 1871 Version of the Theory of Ideals.Jeremy Avigad - manuscript
    By the middle of the nineteenth century, it had become clear to mathematicians that the study of finite field extensions of the rational numbers is indispensable to number theory, even if one’s ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the “integers” in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the ordinary integers. In 1844, Ernst Kummer observed that unique factorization (...)
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  21. Odel and the Metamathematical Tradition.Jeremy Avigad - manuscript
    The metamathematical tradition that developed from Hilbert’s program is based on syntactic characterizations of mathematics and the use of explicit, finitary methods in the metatheory. Although G¨.
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  22. Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals.Jeremy Avigad - manuscript
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation.
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  23. Type Inference in Mathematics.Jeremy Avigad - unknown
    In the theory of programming languages, type inference is the process of inferring the type of an expression automatically, often making use of information from the context in which the expression appears. Such mechanisms turn out to be extremely useful in the practice of interactive theorem proving, whereby users interact with a computational proof assistant to constructformal axiomatic derivations of mathematical theorems. This article explains some of the mechanisms for type inference used by the "Mathematical Components" project, which is working (...)
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  24. Tait William. The Provenance of Pure Reason: Essays in the Philosophy of Mathematics and its History. Oxford University Press, Oxford, 2005, X+ 332 Pp. [REVIEW]Jeremy Avigad - 2006 - Bulletin of Symbolic Logic 12 (4):608-611.
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  25. Mathematical Method and Proof.Jeremy Avigad - 2005 - Synthese 153 (1):105-159.
    On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
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  26. Theorem Proving in Lean.Jeremy Avigad, Leonardo de Moura & Soonho Kong - unknown
    Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point (...)
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  27. Structuralism, Invariance, and Univalence†: Articles.Steve Awodey - 2013 - Philosophia Mathematica 22 (1):1-11.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  28. How to Nominalize Formalism &Dagger.Jody Azzouni - 2005 - Philosophia Mathematica 13 (2):135-159.
    Formalism shares with nominalism a distaste for _abstracta_. But an honest exposition of the former position risks introducing _abstracta_ as the stuff of syntax. This article describes the dangers, and offers a new escape route from platonism for the formalist. It is explained how the needed role of derivations in mathematical practice can be explained, not by a commitment to the derivations themselves, but by the commitment of the mathematician to a practice which is in accord with a theory of (...)
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  29. Mathematics and Explanatory Generality.Alan Baker - forthcoming - Philosophia Mathematica:nkw021.
    According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that (...)
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  30. Critical Studies / Book Reviews. [REVIEW]Mark Balaguer - 1998 - Philosophia Mathematica 6 (3):108-126.
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  31. Armour-Garb, B., 491.A. Baltag, E. C. Banks, L. Boi, G. Bonanno, B. Brogaard, L. K. C. Cheung, D. Costantini, U. Garibaldi, V. Goranko & C. Hitchcock - 2004 - Synthese 139 (515).
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  32. Understanding Thermodynamic Singularities: Phase Transitions, Data, and Phenomena.Sorin Bangu - 2009 - Philosophy of Science 76 (4):488-505.
    According to standard (quantum) statistical mechanics, the phenomenon of a phase transition, as described in classical thermodynamics, cannot be derived unless one assumes that the system under study is infinite. This is naturally puzzling since real systems are composed of a finite number of particles; consequently, a well‐known reaction to this problem was to urge that the thermodynamic definition of phase transitions (in terms of singularities) should not be “taken seriously.” This article takes singularities seriously and analyzes their role by (...)
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  33. Pythagorean Heuristic in Physics.Sorin Bangu - 2006 - Perspectives on Science 14 (4):387-416.
    : Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting to (...)
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  34. Steiner on the Applicability of Mathematics and Naturalism †I Would Like to Thank to Margaret Morrison, Jim Brown, Mark Steiner, Alasdair Urquhart, Patricia Marino, and Two Anonymous Referees of This Journal for Helpful Comments and Discussions.Sorin Bangu - 2006 - Philosophia Mathematica 14 (1):26-43.
    Steiner defines naturalism in opposition to anthropocentrism, the doctrine that the human mind holds a privileged place in the universe. He assumes the anthropocentric nature of mathematics and argues that physicists' employment of mathematically guided strategies in the discovery of quantum mechanics challenges scientists' naturalism. In this paper I show that Steiner's assumption about the anthropocentric character of mathematics is questionable. I draw attention to mathematicians' rejection of what Maddy calls ‘definabilism’, a methodological maxim governing the development of mathematics. I (...)
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  35. In Support of Significant Modernization of Original Mathematical Texts.A. Barabashev - 1997 - Philosophia Mathematica 5 (1):21-41.
    At their extremes, the modernization of ancient mathematical texts leaves nothing of the source and the refusal to modernize changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources in the context of a socio-cultural philosophy of mathematics.
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  36. Chris Pincock , Mathematics and Scientific Representation . Reviewed By.Sam Baron - 2013 - Philosophy in Review 33 (1):63-66.
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  37. Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles.Claudio Bartocci, Renato Betti, Angelo Guerraggio & Roberto Lucchetti - 2011 - Springer.
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  38. A Constructive Theory of Continuous Domains Suitable for Implementation.Andrej Bauer & Iztok Kavkler - 2009 - Annals of Pure and Applied Logic 159 (3):251-267.
    We formulate a predicative, constructive theory of continuous domains whose realizability interpretation gives a practical implementation of continuous ω-chain complete posets and continuous maps between them. We apply the theory to implementation of the interval domain and exact real numbers.
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  39. Two Constructive Embedding-Extension Theorems with Applications to Continuity Principles and to Banach-Mazur Computability.Andrej Bauer & Alex Simpson - 2004 - Mathematical Logic Quarterly 50 (45):351-369.
    We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful (...)
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  40. Arithmetic as Propaedeutic to Theology: The Brethren of Purity.François Beets - 2015 - Balkan Journal of Philosophy 7 (1):71-76.
    In the 10th century, the Brethren of Purity conceived a henological arithmetic which they believed could explain the mathematical structure of the cosmos, and could lead the student to the discovery of the real substance of his own soul, a discovery which is the first step towards knowledge of metaphysical and theological truth.
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  41. A Detail in Kronecker's Program.E. T. Bell - 1936 - Philosophy of Science 3 (2):197-207.
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  42. Finite or Infinite?E. T. Bell - 1934 - Philosophy of Science 1 (1):30-49.
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  43. Gregory H. Moore. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. Mineola, N.Y.: Dover Publications, 2013. ISBN 978-0-48648841-7 . Pp. 448: Critical Studies/Book Reviews. [REVIEW]John L. Bell - 2014 - Philosophia Mathematica 22 (1):131-134.
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  44. Van (2001). The Creative Growth of Mathematics.J. P. Bendegem - 1999 - Philosophica 63:1.
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  45. "Symmetry." By Hermann Weyl [Article]. [REVIEW]J. D. Bernal - 1955 - British Journal for the Philosophy of Science 5 (20):335.
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  46. Genealogical Mathematics.Karl Bernard - 1990
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